Sufficient conditions for smoothing codimension one foliations

Author:
Christopher Ennis

Journal:
Trans. Amer. Math. Soc. **276** (1983), 311-322

MSC:
Primary 57R30; Secondary 57R10, 58F18

MathSciNet review:
684511

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Abstract: Let be a compact manifold. Let be a nonsingular vector field on , having unique integral curves through . For continuous, call whenever defined. Similarly, call .

For , a foliation of is said to be smoothable if there exist a foliation , which approximates , and a homeomorphism such that takes leaves of onto leaves of .

Definition. A transversely oriented Lyapunov foliation is a pair consisting of a codimension one foliation of and a nonsingular, uniquely integrable vector field on , such that there is a covering of by neighborhoods , , on which is described as level sets of continuous functions for which is continuous and strictly positive.

We prove the following theorems.

Theorem 1. *Every* *transversely oriented Lyapunov foliation* *is* *smoothable to a* *transversely oriented Lyapunov foliation* .

Theorem 2. *If* *is a* *transversely oriented Lyapunov foliation, with* *and* *continuous for* *and* , *then* *is* *smoothable to a* *transversely oriented Lyapunov foliation* .

The proofs of the above theorems depend on a fairly deep result in analysis due to F. Wesley Wilson, Jr. With only elementary arguments we obtain the version of Theorem 1.

Theorem 3. *If* *is a* *transversely oriented Lyapunov foliation, with* *and* *is continuous, then* *is* *smoothable to a* *transversely oriented Lyapunov foliation* .

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1983-0684511-4

Article copyright:
© Copyright 1983
American Mathematical Society